{
 "metadata": {
  "name": "",
  "signature": "sha256:54b0ca29b543517953a1bf9cff1ed8eae17d232857488587a9dd3b3a3dab14ca"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from __future__ import print_function, division\n",
      "import thinkbayes2\n",
      "import thinkplot\n",
      "%matplotlib inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 40
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Ignore the first few cells for now -- they are experiments I am working on related to the prior."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "mu = 1\n",
      "pmf = thinkbayes2.MakeExponentialPmf(mu, high=1.0)\n",
      "thinkplot.Pdf(pmf)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Warning: Brewer ran out of colors.\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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BiIigZCAiIigZiIgISgYiIoKSgYiIoGQgIiIoGYiICEoGIiKCkoGIiKBkICIi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\nERERERERqWv/D+3ChglzYzYwAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f19a34019d0>"
       ]
      }
     ],
     "prompt_number": 41
    },
    {
     "cell_type": "raw",
     "metadata": {},
     "source": [
      "Ignore"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "mu = 5\n",
      "pmf = thinkbayes2.MakeExponentialPmf(mu, high=1.0)\n",
      "thinkplot.Pdf(pmf)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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r1u9kw6a9VFZWNztPTEwk/ftkcF6frvTvk0n/3hm6qZ60C4WBiEVqamrZtusg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\nRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f19a3409350>"
       ]
      }
     ],
     "prompt_number": 42
    },
    {
     "cell_type": "raw",
     "metadata": {},
     "source": [
      "Ignore"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "metapmf = thinkbayes2.Pmf()\n",
      "for lam, prob in pmf.Items():\n",
      "    if lam==0: continue\n",
      "    pmf = thinkbayes2.MakeExponentialPmf(lam, high=30)\n",
      "    metapmf[pmf] = prob\n",
      "    \n",
      "interarrival = thinkbayes2.MakeMixture(metapmf)\n",
      "thinkplot.Pdf(interarrival)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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6u/20tHbQ1Oyh6XwHjU1umpo7aGrymPOM58YmT9gf8RtJqamOviGR4SQz88Lg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\n9lmQAogAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f19ccf40510>"
       ]
      }
     ],
     "prompt_number": 43
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Ok, let's start here.  Suppose we know $\\lambda$.  We can compute the distribution of interarrival times (times between logins)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "lam = 0.1    # average arrival rate in logins per day\n",
      "interarrival = pmf = thinkbayes2.MakeExponentialPmf(lam, high=90)\n",
      "thinkplot.Pdf(interarrival)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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B+f18v/oIzlBjSyc//XM1r1fV91m/oLyAf75hLudMGetQZCIST/EYPurBdPi+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\niIiIiEii+f/zR8D3x3Q8cAAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f19a3498e50>"
       ]
      }
     ],
     "prompt_number": 44
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If we observe someone, we are more likely to land during a longer interval."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "observed = interarrival.Copy()\n",
      "for val, prob in observed.Items():\n",
      "    observed[val] *= val\n",
      "observed.Normalize()\n",
      "\n",
      "print(interarrival.Mean(), observed.Mean())\n",
      "thinkplot.Pdf(observed)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "9.76490343621 19.9050636319\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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7t2jI/IJH3j3GO1qPSGTKJUwiUNNQcvr0iorQrmaBYJBv/f4gjZ29DkclkloS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\nw19Dx67QtUyvm62vv8oVV11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       "text": [
        "<matplotlib.figure.Figure at 0x7f19a34e8190>"
       ]
      }
     ],
     "prompt_number": 45
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If we land during an intererval of duration $x$, the time since last login is uniform between 0 and $x$.  So the distribution of time since last login (`timesince`) is a mixture of uniform distributions."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "metapmf = thinkbayes2.Pmf()\n",
      "for time, prob in observed.Items():\n",
      "    if time == 0:\n",
      "        continue\n",
      "    pmf = thinkbayes2.MakeUniformPmf(0, time, 101)\n",
      "    metapmf[pmf] = prob\n",
      "    \n",
      "timesince = thinkbayes2.MakeMixture(metapmf)\n",
      "print(timesince.Mean())\n",
      "thinkplot.Pdf(timesince)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "9.95253181595\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x7f19cdba66d0>"
       ]
      }
     ],
     "prompt_number": 46
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The data is in the form of \"time since last login\", so we need to be able to look up a time, $t$, and get the probability density at $t$.  But we have a PMF with lots of discrete times in it, so we can't just look it up.  One option: Compute the CDF, generate a sample, and estimate the PDF by KDE:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "cdf = thinkbayes2.Cdf(timesince)\n",
      "thinkplot.Cdf(cdf)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 47,
       "text": [
        "{'xscale': 'linear', 'yscale': 'linear'}"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAXUAAAEACAYAAABMEua6AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nXze1Pdy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       "text": [
        "<matplotlib.figure.Figure at 0x7f19a360e950>"
       ]
      }
     ],
     "prompt_number": 47
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Get a sample:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sample = cdf.Sample(10000)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 48
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Estimate the PDF:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "pdf = thinkbayes2.EstimatedPdf(sample)\n",
      "thinkplot.Pdf(pdf)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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h8u4o+fIoeXd01pFA7ZCwbFKJDKlEmrTdRdruIhM8JzTVhCyAhUj4NvAmcC9w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\niIiIiIiIiIiIiIgsfv8fZLH2Cj251wYAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f19cdba6a10>"
       ]
      }
     ],
     "prompt_number": 49
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Second option: use numerical differentiation to compute the derivative of the CDF, which is the PDF:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import scipy\n",
      "import numpy\n",
      "xs = numpy.linspace(0, 90, 101)\n",
      "ys = [scipy.misc.derivative(cdf.Prob, x) for x in xs]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 50
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Numerical differentiation is more accurate, especially near zero.  The value at zero is wrong: there are ways we could fix it, but it's not necessary because we won't get zero as data."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "thinkplot.plot(xs, ys)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Warning: Brewer ran out of colors.\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Xz9369rWxhD59Wm99+9q9Bf36aUxBgleIhF8GbAQuBXYCrwDTgA2eNlOAWc7r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\nAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f19a3602b90>"
       ]
      }
     ],
     "prompt_number": 51
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 51
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": []
    }
   ],
   "metadata": {}
  }
 ]
}